Infinite Wordle and the Mastermind numbers

TL;DR

This paper explores infinite variants of Wordle and Mastermind, revealing that in infinite Wordle the secret can always be guessed in finite steps, while the minimal guesses in infinite Mastermind relate to complex set-theoretic cardinal characteristics.

Contribution

It introduces and analyzes the infinite versions of Wordle and Mastermind, connecting their properties to advanced concepts in set theory and cardinal characteristics.

Findings

Infinite Wordle can be solved in n steps for words of size n.

The mastermind number for infinite sequences is uncountable and independent of ZFC.

The mastermind number equals the covering number of the meager ideal, linking the game to set-theoretic hierarchy.

Abstract

I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game-theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and allowing transfinite game play. For each game, a secret codeword is hidden, which the codebreaker attempts to discover by making a series of guesses and receiving feedback as to their accuracy. In Wordle with words of any size from a finite alphabet of $n$ letters, including infinite words or even uncountable words, the codebreaker can nevertheless always win in $n$ steps. Meanwhile, the mastermind number, defined as the smallest winning set of guesses in infinite Mastermind for sequences of length $\omega$ over a countable set of colors without duplication, is uncountable, but the exact value turns out to be independent of ZFC, for it is provably equal to the eventually different number $\frak{d}({\neq^*})$, which is the same as the covering number of the meager ideal $\text{cov}(\mathcal{M})$. I thus place all the various mastermind numbers, defined for the natural variations of the game, into the hierarchy of cardinal characteristics of the continuum.